Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-07-06T04:39:03.492Z Has data issue: false hasContentIssue false
  • English
  • Français

S-strictly quasi-concave vector maximisation

Published online by Cambridge University Press:  17 April 2009

Hong-Bin Dong ,
Xun-Hua Gong ,
Shou-Yang Wang  and
Luis Coladas
Hong-Bin Dong
Affiliation:
Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China, e-mail: donghb@amss.ac.cn
Xun-Hua Gong
Affiliation:
Department of Mathematics, Nanchang University, Nanchang 330047, China e-mail: ncxhgong@263.ne
Shou-Yang Wang
Affiliation:
Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China e-mail: sywang@iss02.iss.ac.cnSchool of Business Administration, Hunan University, Changsha, Hunan 410082, China e-mail: sywang@hnu.edu.cn
Luis Coladas
Affiliation:
Department of Statistics and Operations Research, Universidade de Santiago, 15782 Santiago de Compostela, Spain, e-mail: coladas@usc.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we discuss the relationship among the concepts of an S-strictly quasiconcave vector-valued function introduced by Benson and Sun, a C-strongly quasiconcave vector-valued function and a C-strictly quasiconcave vector-valued function in a topological vector space with a lattice ordering. We generalize a main result obtained by Benson and Sun about the closedness of an efficient solution set in multiple objective programming. We prove that an efficient solution set is closed and connected when the objective function is a continuous S-strictly quasiconcave vector-valued function, the objective space is a topological vector lattice and the ordering cone has a nonempty interior.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 67 , Issue 3 , June 2003 , pp. 429 - 443
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Benoist, J., ‘Connectedness of the efficient set for strictly quasiconcave sets’, J. Optim. Theory Appl. 96 (1998), 627–654.CrossRefGoogle Scholar
[2]Benson, H.P. and Sun, E.J., ‘New closedness results for efficient sets in multiple objective mathematical programming’, J. Math. Anal. App. 238 (1999), 277–296.CrossRefGoogle Scholar
[3]Choo, E.U., Schaible, S. and Chew, K.P., ‘Connectedness of the efficient set in three-criteria quasiconcave programming’, Cahiers Centre Études Rech. Opér. 27 (1985), 213–220.Google Scholar
[4]Daniilidis, A., Hadjisavvas, N. and Schaible, S., ‘Connectedness of the efficient set for three objective quasiconcave maximization problems’, J. Optim. Theory Appl. 93 (1997), 517–524.CrossRefGoogle Scholar
[5]Fu, W.T., ‘On the closedness and connectedness of the set of efficient points’, (in Chinese), J. Math. Res. Exposition 14 (1994), 89–96.Google Scholar
[6]Fu, W.T. and Zhou, K.P., ‘Connectedness of the efficient solution sets for a strictly path quasiconvex programming problem’, Nonlinear Anal. 21 (1993), 903–910.Google Scholar
[7]Fu, W.T. and Zhou, K.P., ‘The connectedness of the efficient solution sets for a strongly quasiconvex multiobjective optimization in infinite dimensional spaces’, (in Chinese), Acta Math. App. Sinica 18 (1995), 140–146.Google Scholar
[8]Hu, Y.D. and Sun, E.J., ‘Connectedness of the efficient set in strictly quasiconcave vector maximization’, J. Optim. Theory Appl. 78 (1993), 613–622.CrossRefGoogle Scholar
[9]Jameson, G., Ordered linear spaces, Lecture Notes in Mathematics 141 (Springer-Verlag, Berlin, 1970).CrossRefGoogle Scholar
[10]Schaefer, H.H., Topological vector spaces (Springer-Verlag, New York, 1970).Google Scholar
[11]Schaible, S., ‘Bicriteria quasiconcave programs’, Cahiers Centre Études Rech. Opér. 25 (1983), 93–101.Google Scholar
[12]Sun, E.J., ‘On the connectedness of the efficient set for strictly quasiconvex vector minimization problems’, J. Optim. Theory Appl. 89 (1996), 475–481.CrossRefGoogle Scholar
[13]Yu, P.L. and Zeleny, M., ‘The set of all nondominated solutons in linear cases and a multicriteria simplex method’, J. Math. Anal. Appl. 49 (1975), 430–468.CrossRefGoogle Scholar